Gorenstein ring

Remarks

A Gorenstein local ring is Cohen–Macaulay, and then $\operatorname{id}_R R = \dim R$. Equivalently, a Noetherian local ring $(R, \mathfrak{m}, k)$ of dimension $d$ is Gorenstein if and only if $\operatorname{Ext}^i_R(k, R) = 0$ for $i \ne d$ and $\operatorname{Ext}^d_R(k, R) \cong k$. For an Artinian local ring this says that the socle $(0 :_R \mathfrak{m})$ is a $k$-vector space of dimension $1$, or that $R$ is the injective hull of $k$ as an $R$-module. Example: $R = K[x]/(x^n)$ with $n \ge 1$ and $K$ a field is Gorenstein, since its socle is $K x^{n-1}$. Non-example: $R = K[x,y]/(x^2, xy, y^2)$ is Artinian local with socle $Kx \oplus Ky$ of dimension $2$, so it is not Gorenstein, although it is Cohen–Macaulay.